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Specimen paper for entry into Year 12

Maths

Name:……………………………………………

School:……………………………………………

Answer all questions in Section A and either question 13 or 14 in Section B.

Do all your workings in the spaces provided.

Marks for Section A: 60 Marks for Section B: 20

Time allowed : 75 minutes.

You are allowed to use a calculator in this exam. A list of useful formulae will be found on page 2.

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Useful Formulae

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Section A

1. Solve the simultaneous equations

…………………….

……………………. (Total 3 marks)

2. Jothi bought a car. Later he sold the car for £2125. He made a loss of 15%. Work out the original price of the car.

£ ……………………. (Total 3 marks)

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3. (a) Expand and simplify ( )( )

…………………………. (2)

(b) Simplify

………………………… (2)

(c) Factorise

……………………………. (1)

(d) Factorise completely ( )

………………………….. (2)

(e) Factorise

……………………….. (2)

(Total 9 marks)

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4.

ABC is a triangle. D is a point on AB. CD is perpendicular to AB. AD = 7.2 cm, DB = 3.9 cm, AC = 8.3 cm. Calculate the size of angle DBC. Give your answer correct to 1 decimal place.

…………….° (Total 5 marks)

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Diagram NOT drawn accurately

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5. Each time Jeni plays a computer game the probability that she will win is

Jeni plays the computer game 3 times.

Calculate the probability that Jeni will win

(a) all 3 games,

…………………………… (2)

(b) exactly 2 out of the 3 games.

………………………… (3)

(Total 5 marks)

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6.

(a) (i) Calculate the size of angle AOC.

…………°

(ii) Give a reason for your answer.

………………………………………………………………………………………

………………………………………………………………………………………

(2)

(b) (i) Calculate the size of angle ADC.

………...°

(ii) Give a reason for your answer.

……………………………………………………………………………………...

……………………………………………………………………………………...

(2)

(Total 4 marks)

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Diagram NOT drawn accurately

A, B, C and D are points on a circle, centre O. Angle ABC = 58°.

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7. The diagram shows the position of two ships, A and B, and a lighthouse L.

Ship A is 5 km from L on a bearing of 070° from L. Ship B is 3 km from L on a bearing of 210° from L. Calculate the distance between ship A and ship B. Give your answer to 3 significant figures.

…………………. km (Total 3 marks)

Diagram NOT drawn accurately

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8. In the diagram, a sector of a circle of radius 12cm is shaded. The area of the sector is 112π cm2. Calculate the value of x.

………………. (Total 4 marks)

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Diagram NOT drawn accurately

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9. (a) Simplify

……………………. (2)

(b) Simplify

…………………. (4)

(c) Make the subject of the formula ( )

( )

…………………….. (4)

(Total 10 marks)

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10.

A, B and L are points on a circle, centre O. AB is a chord of the circle. M is the midpoint of AB. LOM is a straight line. AB = 24 cm. LM = 18 cm. Calculate the diameter of the circle.

……………… cm (Total 4 marks)

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11. Solve the simultaneous equations

………………………………………………………………….. (Total 6 marks)

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12. (a) (√ ) √ , where

Find the value of .

……………… (2)

(b) Express

√ as a power of 2

……………………….. (2)

(Total 4 marks)

Turn over

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Section B

Answer EITHER question 13 on pages 14, 15 and 16 OR question 14 on page 17. Both questions are worth 20 marks.

13. (a) Find the equation of the perpendicular bisector of the line segment joining the points A (-1, 1) and B (3, 9).

………………………….. (4)

(b) If , find

and hence find the equation of the tangent to the

curve at the point on the curve where .

……………………… (5)

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(c) (i) Express in the form ( ) .

……………………..

(2) (ii) State the coordinates of the turning point

on the graph of .

……………………… (2)

(d) You are given ( ) .

(i) Given that ( ) , express ( ) in fully factorised form.

( ) ……………………………………………. (4)

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(ii) Show that ( ) .

(3) (Total 20 marks)

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14. A small goat is tethered by a rope to a point at ground level on a side of a square barn which is in a large horizontal field of grass. The sides of the barn are of

length and the rope is of length . Let be the area of the grass that the

goat can graze. Prove that and determine the minimum value of . (Total 20 marks)

End of Questions

Question 14 is the final question. Use as much of the remaining space as you need for your solution to Question 14.

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